Post on 30-Nov-2021
NASA Contractor Report 191546/ o2o G
On the Use of the Noncentral
Chi-Square Density Functionfor the Distribution of
Helicopter Spectral Estimates
Donald P. Garber
Lockheed Engineering & Sciences Company
Hampton, VA 23666-1339
Contract NAS 1-19000
October 1993
National Aeronautics andSpace Administration
Langley Research CenterHampton, Virginia 23681-0001
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Contents
Abstract ..................................... 1
Symbol Lilt ................................... 1
Introductien ................................... 2
Mathematical Model ............................... 2
Noncentral Chi-Squvre Distribution ........................ 2
Gaussian Approximation ............................. 4
Confidence Limits ................................ 5
Comparison cf Model with Data .......................... 7
Spectlal Estimates ............................... 7
Synthesized Data ................................ 9
Tiltrotor Hover Data .............................. 9
Helicopter Approach Data ........................... 10
Summary and Conclusions ............................ 11
Acknowledgements ............................... 12
References ................................... 12
Al_pendix A: Mathematical Model ........................ 14
Single Spectlal Estimate ............................ 14
Appendix B: Numerical Methods ......................... 17
Density Function E_aluation .......................... 17
Integration .................................. 18
Root Location ................................ 19
Appendix C: FORTRAN Program ........................ 22
Abstract
A probability density function for the variability ofensemble averaged spectral estimates from helicopter
acoustic signals in Gaussian background noise was eval-uated. Numerical methods for calculating the density
function and for determining confidence limits were ex-
plored. Density functions were predicted for both syn-
thesized and experimental data and compared with ob-
served spectral estimate variability.
Symbol List
A(0
am
c(0
Ell
f0
F
gO
G
H+
L
M
P
P
q
Q
,(t)
R
s(_)
o(,)
envelope function
even Fourier coefficient
odd Fourier coefficient
band-limited Gaussian compositefunction
expectation operator
chi-square limit function
window correction component
noncentral chi-square limit function
window correction component
window correction component
modified Bessel function of order t,
lower confidence limit
Fourier index
number of Gauss-Legendre weightsand abscissas
band-limited Gaussian noise function
number of spectra in ensemble aver-
age
probability density function
amplitude of sinusoidal signal
indexfor statistical moments
window bias function
band-limited Gaussian signal function
ratio of sinusoidal energy to Gaussian
energy
spectral density function
periodic signal function
time
T
_j
U
V
wj
w(,)
wi
W
xo(0
Xb(*)
.(,)
*o(,)
.b(t)
Y
*(*)
t_
7
6i
f
(
'7
0
#
/.2
finite Fourier transform integrationtime
Gauss-Legendre abscissas
upper confidence limit
relative variability
Gauss-Legendre weights
window function
window coeffcients
confidence coefficient
even signal-plus-noise modulationfunction
odd signal-plus-noise modulationfunction
input function to energy detection
system
even noise modulation function
odd noise modulation function
ensemble averaged spectral estimate
normalized ensemble averaged spec-tral estimate
lower limit of integration
upper limit of integration
normalized Bessel function argument
output function from energy detection
system
dummy parameter of degeneratehypergeometric function
area under tail of probability densityfunction
dummy parameter of degenerate
hypergeometric function
improvement to approximate root
frequency mis-match index
dummy argument of degeneratehypergeometrlc function
dummy parameter of chi-equare limltfunction
dummy variable of integration
pure tone function
mean
normalized mean
2nd central moment
_2 normalised 2_ centralmoment
argument of chi-squarelimitfunction
P
_2
T
xt
0
parameter for variety of functions and
equations
energy of random process or sinu-
soidal signal
approximation to inverse standardnormal distribution
time interval
phase function
degenerate hypergeometric function
chi-square distribution
noncentral chi-square distribution
uniformly distributed phase randomvariable
radial frequency
.............. integration operator
Subscripts:
I
n
r
s
filtered function
narrow-band Gaussian noise
narrow-band Gaussian signal
sinusoidal signal
random variable associated with
function
Introduction
The variability of a single spectral estimate for atime series consisting of only band-limited Gaussian
noise has been shown to follow a chi-square probability
density function with two degrees of freedom [1-2]. Ithas also been shown that the variability is independent
of the length of time over which data is collected.
Increasing the temporal length of data improves the
spectral resolution and reduces the bias of the estimate,
but does not reduce the uncertainty inherent in a singlespectral estimate. The method generally employed to
reduce variability is to collect an ensemble of spectral
estimates and then average the ensemble on an energy
or mean power basis. The variability of the ensemble
average is then given by a chi-square density function
with 2N degrees of freedom where N is the number of
independent spectra in the ensemble.
When the time series for which a spectral estimate is
desired substantially violates the assumption of Gaus-
sian variability, Using the chisquare density function to
describe spectral uncertainty can be misleading. The
acoustic signal generated by a helicopter in flight con-tains periodic impulsive noise that gives rise to harmon-
ics of the rotor blade passage frequency in the power
spectrum. Spectral estimates of helicopter noise at fre-
quencies coinciding with blade passage harmonics can
show significantly different variability than that pre-
dicted by a chi-square function [3]. Quite simply, theperiodic impulsive noise does not follow a Gaussian dis-
tribution and, hence, a chi-square density is inadequate
to describe the statistical behavior of its spectral esti-mate.
The purpose of this paper is to show that the math-
ematical methods necessary to describe the effect of pe-
riodic impulsive components on the variability of spec-tral estimates already exist in the literature in books
by Davenport and Root [4], Whalen [5], and Burdic[6], as well as in a variety of scientific papers associated
with signal detection methods [7-9]. Those methods arediscussed very briefly in this paper and are presented
in more detail in Appendix A. A Gaussian approxima-
tion to the resultant statistical function is also provided.Numerical methods for evaluating the statistical func-
tions, for integrating the functions, and for determiningconfidence limits are presented in Appendix B. A FOR-
TRAN 77 computer program that evaluates theoretical
confidence limits for chi-square distributions is listed inAppendix C.
Finally, spectral estimate techniques will be dis-
cussed briefly and the mathematical model will be com-
pared with spectral estimates of real and synthesiseddata to examine its validity. Synthesised data wUl be
used to most accurately represent the assumptions and
approximations made in the development of the model
and to permit an examination of the effects of spectral
bias. Comparison with real data wili include recordings
of tiltrotor hovers at short range and helicopter longrange approach flights.
Mathematical Model
Noncentral Chi-Square Distribution
Power spectra may be obtained from time series data
using any of three equivalent methods [2]. One ap-proach is to perform a Fourier transform on the autocor-
relation of the time series by the Blackman-Tukey pro-
cedure. Another is to directly calculate a finite Fourier
transform of the original time series data. A third way
to obtain a power spectrum, and the approach to bemodeled here, is to process the time series data with an
energy detection system [10] composed of three com-ponents that operate on the data in series as shown in
Figure 1. The first component of the system is a narrow
band-pass filter centered on the frequency of interest.
The second component is a square-law detector that
squares the output of the filter. The output of this
detector is proportional to the instantaneous power in
the filter band. The third component is an averager or
integrator that is equivalent to a low-pass filter. The in-
tegrator converts the instantaneous power output from
the square-law detector into a time-averaged or meanpower that is equivalent to a spectral estimate in the
frequency band over the time of integration.
x(t) -_ Band-Pass lJ Square-LawL.]Filter ] J Detector ] [ Integrator _ /(t)
Figure 1. - Energy detector equiva]ent of spec-tral estimate.
Correspondence of the energy detection system
spectral method with the other spectral methods re-quires additional constraints. The first is that the width
of the band-pass filter must be small compared to thecenter frequency of the filter. The second constraint
is that the width of the low-pass filter must be equal
to the width of the band-pass filter. If the integration
time of the low-pass filter (which is the inverse of the
filter width) is equal to the time length of data usedwith the other spectral methods, the spectral estimates
are equivalent. As will be seen later, the integrationtime will be allowed to increase without limit for the
purposes of the mathematical development. One con-
sequence of this is the elimination of bias from the den-
sity functions derived herein. Obviously, the practicalcalculation of spectral estimates requires a finite data
set, but the mathematical ploy is intended to make the
derivation tractable and will be tolerated for that pur-
pose. The issue of bias will be raised again, briefly, inthe discussion of spectral estimates in the data compar-ison section.
When analyzing acoustic data from wind tunnel and
outdoor experiments involving aircraft, it is convenientto distinguish between that part of the acoustic data
arising from the operation of the aircraft and the back-
ground, or ambient, part. The background will be re-ferred to as noise while that from the aircraft will be
referred to as signal. The signal is further assumed to
consist of a periodic and a random part. Certain ide-
alizations of system behavior and acoustic signal and
noise characteristics are made to simplify the analysis.
The background noise is assumed to be a band-limited
Gaussian process and the signal is assumed to be the
sum of a periodic part and a band-limited Gaussianprocess that is independent of the noise process. The
methods described by Davenport and Root [4] are uti-
lised in the ensuing development. The input function
to the energy detection system z(1) is given by the sumof the signal and noise as
"(0 = si0 + + ,,(0
where sit ) is the periodic part of the signal, r(g) is the
Gaussian part of the signal and nit ) is the noise.The band-pass filter is assumed to pass all frequency
components in the band without distortion and reject
all other frequencies perfectly. The output from the
filter z/(1) is then given by
"t(0 = ss( ) + "t(0 + ,*j(0
where s/(t ) is a constant-amplitude sine wave with a
constant frequency in the pass band, _'j(t) is a narrow-band Gaussian random process associated with the sig-
nal, and n! (_) is a narrow-band Gaussian random pro-cess associated with the background noise. Because the
signal and noise random processes are independent, it
is possible to define a composite narrow-band Gaussian
random process
csiO- 'v(O + "l(t)
where the variance of the composite process, cry, isequal to the sum of the variances of the narrow-band
processes associated with the signal, _, and the back-
ground noise, _,_. For consistency of notation the en-
ergy associated with the sinusoidal part of the signal
will be referred to as _ = P2/2 where P is a constantamplitude.
The integrator is assumed to be an ideal low-pass
filter that passes all frequency components in the band
without distortion and rejects all other frequencies per-
fectly. The output from the integrator and, hence, from
the energy detection system is given by z(t). The prob-
ability density function of a sample of the output, zt,
can be written (see Appendix A for derivation)
P(Zd = ( _ ) e-[*'+_3"]/_ l° f 2cr'--v/__\cr_ }
which is referred to as a noncentral chi-square distribu-tion [5].: This probability density function describes the
variability inherent in a single spectral estimate when
the time series contains a periodic part that is man-
ifested as a harmonic in the frequency band. When
there is no periodic part contained in the time series or
when the frequency band under consideration does not
include a harmonic of the periodic signal, the density
function is simply
which is an exponential function or, equivalently, achi-square function of two degree s 0f freedom. These
density functions exactly correspond to results obtained
by Burdic [6] for a matched filter detector in an active
pulse-echo detection system.
When an ensemble of N independent samples of the
detection system output is averaged, the new randomvariable
1 N
/=1
has a probability density function that can be ob-tained by analogy with the results for the multiple pulse
matched filter detector given by Burdic [6] or by coor-dinate transformation of the normalized distribution of
Whalen [5]
N y e_N(_+_:.)#,_IN__\ _rc /
This probability density function describes the variabil-
ity inherent in an ensemble averaged spectral estimate
when the time series contains a periodic part tha t ismanifested _ a harmonicin the frequency band. When
there is no periodic part contained in the time series or
when the frequency band under consideration does not
include a harmonic of the periodic signal, the density
function is simply
which is a chi-square function of 2N degrees of freedom.An interesting feature of this equation is that when
there is no random component of acoustic signal in the
time series, the _ term can simply be replaced by _2n.
Consequently, the origins of the random components ofthe time series, at least insofar as they are Gaussian
and independent, are unimportant to the shape of the
density function. It is also clear that where random
Gaussian components alone are present, their relative
levels are unimportant and the absolute level has no
effect on the relative variability about the mean.
Gaussian Approximation
When the number of spectra included in the averageis large, the Central Limit Theorem can be invoked
[11]. to approximate the noncentral chi-square function
with a Gaussian function. To do this, however, a meanand variance are required. The qth moment of the
noncentral chi-square distribution is given by
g[y ] = f0=
L J x o-o /
where the degenerate hypergeometric function _ is
given by [13].
q,(a, 'y; () = 1 + ( + _- \_--_._..1 +...
The mean of an ensemble average spectral estimate istherefore
which indicates that, as mentioned in the introduction,allowing the integration length to grow without limit
has allowed the derivation of a probability density func-tion tha_ s_o_s no effecl;of the-bfaa-_ssociate_d-w_h-fi-
nite data lengths. The variance of an ensemble averageis given by
= g[:] - u2= +
from which a Gaussian probability density function cannow be written _
p(y) _ --_e-(_-_')_l_'_
that approximates the noncentral chi-square function
for large sample size or when the ratio of periodic
energy to Gaussian energy in the spectral band is great.Two parameters useful for quantifying the deviation of
a statistical distribution from a Gaussian shape and,hence, the applicability of a Gaussian approximationare skewness
2[ 1+3R]*n = _ (I+ 2R)S/aJ
and (excess) Kurtosis
6[ ,+4R1L({
where R 2 2= #,/#c is the ratio of periodic or tonal energy
in the spectral band to the total Oaussian energy (signaland noise) in the band.
By defining a variable that quantifies the relative
variability of a spectral estimate about the expectedvalue
and solving for the lower confidence limit, L, and the
upper confidence limit, U, where W = 1 - 23 is theconfidence coefficient for a two-tailed interval. When
there is no sinusoidal component in a spectral estimate,the density function is chi-square with 2N degrees of
freedom and it can be shown that the limits are given
by L -- $,/N where _ is determined by solving
N-1 _kiT.'k=O
= 2 c#. / l+2RV = _ / #4 + #_ :I/
it is possible to show that when no periodic componentsare present in the time series, then
and U = _/N where _ is determined by solving
N-1 ek
k=O
which serves to reiterate the point made above. When
there is a periodic component to the time series that is
substantially greater than the total random component
the relative variability is approximately
and the relative variability decreases as the ratio of
tonal energy to random energy increases. Clearly, the
relative variability is always less when there is a tonal
component present than when there is not. If the ratio
of tonal energy to Gaussian energy increases withoutbound or if the number of samples in the ensemble av-
erage increases without bound, both skewness and (ex-cess) Kurtosis approach zero and the Gaussian approx-
imation approaches the noncentral chi-square densityfunction
Confidence Limits
The confidence limits of a density function defined
only for positive arguments are determined by integrat-ing under the tails
and either
or_o U1 - 3 = v( )ay
The expected value of the normalized density function
for noise only is just unity and the variance is givenby 1/I'I. When a sinusoidal component is present in
a spectral estimate, the density function is noncentral
chi-square with 2N degrees of freedom and numericalintegration must be performed. The details of the
numerical integration scheme are given in AppendixB. Solving for the upper and lower confidence limits
involves finding the root of each of the integrations.The numerical methods used for this process are also
detailed in Appendix B.
A comparison of the noncentral chi-square, X_,and chi-square, X02, distributions is shown in Pigure 2below. For this example, both distributions have four
degrees of freedom because they describe an averageof two spectra. Both distributions have the same
mean (or total energy) but the ratio of tonal energy
to broadband energy is R = 10 (or 10 dB) for thenoncentral distribution. The horizontal scale of the
plot is normalized by the noise energy of the noncentral
distribution so the total energy of both distributions is
given by 1 + R. The noncentral distribution, denotedby the solid line, is noticably narrower than the central
distribution, denoted by the dashed line.
The upper and lower 80% confidence limits, ex-
pressed in decibels, are asymmetrically spaced becauseof both the decibel scale and the asymmetry of the dis-tributions. The plot shows there is an 80% confidence
that the actual spectral level is no more than 1.43 dBabove and no more than 1.94 dB below the estimated
level when the tone-to-noise ratio is 10 dB in the spec-tral band. This compares with an 80% confidence in-
terval from 2.89 dB above to 5.75 dB below an estimate
in a spectral band containing only broadband noise. As
the tone-to-noise ratio approaches sero the shape of
the noncentral chi-square distribution approaches the
shape of the chi-square distribution. As the tone-to-noise ratio increases the noncentral chi-square distri-
bution becomes narrower and the confidence limits ap-proach the estimated spectral level.
Number of Spectra: 2
Tone-to- Noise Ratio: I0.00 dB
Confidence Coefficient: 80.00
-- Xt z, Upper LimH: 1.43 dE
.10 - XRz, Lower Limit: -1,94 dE
"-- Xo=, Upper Limit: 2.89 dB_" Xo, Lower Limit: -5.75 dS
rt J ._
-y ",,, \ i
O. 0 10 20 30 40 50
Z
Figure 2. - Comparison of noncentral chl-
square, X_, and chi-square, X_, distributions.
A comparison of the noncentral chi-square distribu-
tion, X_, and a Gaussian approximation is shown inFigure 3 below. The noncentral chi-square distribution
again has four degrees of freedom because it describes
an average of two spectra. The tone-to-noise ratio is5 dB so R _, 3.16. The Gaussian distribution has the
same mean (or total energy) as the noncentral distribu-tion. The horizontal scale of the plot is again normal-ized by the noise energy of the noncentral distribution
so the total energy of both distributions is again givenby 1 + R. The variance of the Gaussian distribution is
given by (1 + 2R)/N. The noncentral distribution, de-noted by the solid line, is noticably more skewed than
the Gaussian distribution, denoted by the dashed line,
which is symmetric.
.25 ' ' ' _ I ' _ ' ' I ' * ' ' l ' ' ' '
Number of Spectra: 2
Tone-to-Noise Ratio: 5.00 HB
.20 / ,"'_" Confidence Coefflclenh 99,90 _;
/ ' _ "_ -- Xa z, Upper Lira;i: 4,86 dB
// \\ X,z, Low.rLimlt: -11.84dB
.15 // \', --- Gousslon Upper Limit: 4.00dB
,'_ 1/ \',, Co...,onLo,,.erLira.:
a. , ,.10 , ,
.05
o.o_L.x ..... I , ,,-'_-.'L'C--L_ l, , I , , , ,5 10 15 20
Z
Figure 3. - Comparison of noncentral chi-
square, X._, and Gaussian distributions.
m
o
.r-
E°m
4DO¢-4D
"10oD
t-O
O
The plot shows there is 99.9% confidence that the
actual spectral level is no more than 4.86 dB above and
no more than 11.84 dB below the estimated level when
the tone-to-noise ratio is 5 dB in the spectral band.
The Gaussian distribution has an upper 99.9% confi-dence limit of 4 dB but an undefined lower confidence
limit because the lower tail of the curve extends below
zero. As either the tone-to-noise ratio or the number of
spectra increases the Gaussian distribution more closelyapproximates the noncentral chi-square.
A comparison of the 80% confidence limits from the
noncentral chi-square distribution, X_, and chi-square,X02, distribution is shown in Figure 4 with a tone-to-
noise ratio for the noncentral distribution of R = 10
(or 10 dB). As the number of spectra included in aspectral estimate increases the confidence limits of both
distributions tend to converge toward the estimate. The
limits of the noncentral distribution, denoted by thesolid line, are always within the limits of the centraldistribution, denoted by the dashed line.
4
2
0
-2
-4
coo,,o.::.oo.:,,:,::,::o.oo,.:-6 _ _No_c.ontral Chl;Square. Xa= ---
- I 00 20 40 60 so _coNumber of spectra
Figure 4. - Comparison of confidence limits from
noneentral chi-square, X_, and chi-square, X0z,distributions.
A comparison of the 99.9% confidence limits from
the noncentral chi-square distribution, X_, and Gaus-sian approximation is shown in Figure 5 with a tone-
to-noise ratio of R _ 3.16 (or 5 dB). As the numberof spectra included in a spectral estimate increases the
confidence limits of both distributions tend to convergetoward the estimate and towards each other. The limits
of the noncentral distribution, denoted by the solid line,
are always above the limits of the central distribution,denoted by the dashed line.
A comparison of the 90% confidence limits from the
noncentral chi-square distribution, X_, its Gaussian
approximation, the chi-square distribution, )Ca, and itsGaussian approximation is shown in Figure 6 with a
tone-to-noise ratio of R _. 3.16 (or 5 dB). The Gaussian
6
approximations converge on their respective chi-squaredistributions as the number of spectra included in an
average increases. The convergence tends to be more
rapid for the noncentral chi-square distribution and itsapproximation.
10 '''r
_ 0 -
E-- j,
-I0 lOE
E -20o i
El
-U 0
d
B
-5OE
"13
C -100
¢J
' ' ' I ' ' ' I ' ' ' t ' ' '
Tone-to-Noise Ratio: 5.00 dB
Confidence Coefflclont: 99.90 g
Noncenlrol ChI-Square, Xt=
....... Gaussian Approximation !o X_=
_300 h , , I , r , l , , , I , i I l20 40 SO 80
Number of spectra
I I
100
Figure 5. - Comparison of confidence limits from
noneentral chi-square, X_, and Gaussian ap-
proximation to X_-
,_I I l I I llllJ I I I l llllJ I 1_a,,.+ -_ J
/,/ .,
/" / Tone-to-Noise Ratio: 5.00 dS' : Confidence Coefficient: 90.00 _{
,/ // ; -- Noncenlrol Chl-$quaro, X. z
/ ..... Gousslon Approximolion to Xm=/'
] .... Chl-Squore, Xoa
.... Goussian Approximation to Xo=
--1_1 J J _ i JlllJ10 w I I I i J,,,Jlo z i i i ],_LtG
Number of spectra
Figure 6. - Comparison of confidence limits fromvarious distributions.
Graphs of 80% and 90% confidence limits are shown
in Figures 12 and 13, respectively, for tone-to-noise
ratios of-co, 0, 5, 10, and 20 dB and for number of
spectra ranging from 1 to 25.
Comparison of Model with Data
Spectral Estimates
The development of the noncentral chi-square dis-
tribution depends on the assumption that the inte-gration time of a Fourier transform is allowed to in-
crease without limit. Practical evaluation of spectral
estimates generally entails the Fast Fourier Transform(FFT) which is a discrete finite transform. Both the
discretization process and the finite length data recordintroduce features to spectral estimates that are not
considered in the development of the density function
presented here. Since the model is to be compared withdata that is necessarily finite, the effect of bias on a
spectral estimate that is introduced by a finite trans-
form must be considered. The bias in a spectral esti-
mate of random noise has been shown to be roughlyproportional to the second derivative of the spectrum
and inversely proportional to the square of the period
of integration [1]. For this derivation, though, only the
bias introduced by a finite transform on a pure tone isconsidered.
Let a pure tone of frequency Wo be represented by
O(t) = sin(wot + _b)
where ¢ is a random phase distributed uniformly on[-_r,_r}. The magnitude of a spectral estimate at wecan then be written as
Q(W)= IO( e)l2
where
2/;= w(t) (t)cos( eOdt
i2_IT w(_),_(_)sin(_et)dt+ TJo
is the finite Fourier transform of the windowed tone
and w(t) is the window function. The expected valueof Q(¢) is then given by
1 f'w
Q = Q(¢)d¢
For a Dirichlet or rectangle window, w(t) = 1, with
ate = wo the expected value of the magnitude is Q = 1
indicating no bias. The expected value of the magnitudeof a spectral estimate for a windowed pure tone of a
frequency different from the estimate frequency is thenan estimate of bias introduced by the frequency mis-match and finite window length.
For a discrete transform the spectral estimates are
made only at discrete frequencies
+.,, = kAw
where A_a = 27c/T is the resolution of an FFT, T is
the integration time or length of a data record, and k
is a frequency index. The tone frequency may then bespecified as
.,o = (k+ Aw
where k is the index that refers to the discrete frequency
nearest the tone frequency and e = (wo- we)/Aw isan indicial measure of the difference between tone and
estimate frequencies such that
1 1--<¢< -
2- -2
A number of common window functions [12] may berepresented by
w(0 = wo - Wlcos(Aw )+ w2 cos (2Awt)
- ws cos (3Awt)
where various choices for the coefficients wl define par-ticular windows. The coefficients may be chosen so thatthe window is power preserving insofar as the total en-
ergy or mean power of a spectral estimate of a random
process is unbiased on the average
w'(t)dt : 1
Coefficients of a number of window functions of this
type [I2] expressed in a power preserving form areshown in Table 1 below.
Table I. - Coefficients of selected windows.
Window w0 wl w 2 wa
Dirichlet 1,0000 0.0 0.0 0.0
Hanning 0.8165 0.8165 0.0 0.0
Hamming 0.8566 0.7297 0.0 0.0
Blackman 0.7610 0.9060 0.1450 0.0
Kaiser-Bessel 0.7463 0.9236 0.1823
4-sample, a = 3
0.0226
Blackman-Harris 0.7063 0.9614 0.2782 0.0230
min 4-sample
The bias correction may now be written in terms of
the window coefficients, wl, estimate frequency index,k, and mis-match index, ¢, as
with
Q = sinc_ (Tre) ( F2 +2 G2)
and
2k + 2_'_F = w0 Tk
G= wo +_H-
Hi=-wl _+(2k+_)a_l
+ w_ + (2k +_)2 _ 4
- w._ + (2k + _)2 _ 9
For the case where the tone frequency and estimate
frequency are the same, We = Wo or • = 0, a window
introduces some bias with the correction given, forexample, by
201oglo[Q]= 201og10[w0]
-1.76 dB for Hanning
-1.34 dB for Hamming
In other words, the discrete finite transform spectralestimate of a pure tone with a frequency other than one
of the discrete estimate frequencies will exhibit a peak
lower then the magnitude of the tonal energy (bias)and will show the remaining energy from that tonespread across every discrete frequency of the estimate
(leakage). The value Q gives the fraction of the originaltonal energy that resides in the frequency bin we nearest
the tone frequency wo. This fraction of the originaltonal energy that remains in the spectral frequency bin
is the value that should be used for the energy of the
tonal signal, Q_r_ rather than o,_, in calculating thedensity function and evaluating confidence limits.
The bias correction may be made for any arbitrarysymmetric window in the same manner. The window
correction component H:E may be expressed as
where the window coefficients, wl, are determined bydiscrete Fourier transform of the window.
Synthesized Data
Synthesized data were used to verify the mathemat-
ical form of the noncentral chi-square density function
and the numerical methods employed in its evaluation.
Data records were constructed by adding white noiseto sine waves. Each value of white noise for each data
record was generated as an independent sample from a
standard normal distribution by a commercial pseudo-
random number generator and then scaled to the ap-propriate absolute level. The phase of the sine wave
for each data record was generated as an independent
sample from a uniform distribution on the interval C0,1)
by a commercial pseudorandom number generator and
then transformed to the interval (-Tr, _r). Each datarecord was transformed by an FFT after a window was
applied and the squared magnitude was then calculated
to generate a spectral record. A spectral average wasdetermined from the appropriate number of spectralrecords and the resulting spectral estimate at the fre-
quency of interest was recorded for statistical analysis.
s.0 ' I ' I ' I ' t ' I ' I ' I ' I ' JN " Tone Level: 70.00 dg ]
C 7.0 - Tone Frequency: 20,00 Hz -'1
>" " No;se Level: 59.90 dE j"[3
s.O - r_ stzs: 2og8 •1: " R..ol-,O.: ,.,gH_ 1O 5.0 - _ Wlndow: g-sample Kolser-Besssl-
" - _ Spectra: 2 -t-
O.O- /\ o_o. ,ooo_3.0
2,0 - _ .AI "',, -- S_os.d _
g l.O
O.o.];_, I.-r'"i , 1 , I_'_4--_ L, "]'"i"'| ......lo .20 .so .go .50 so ,7o .so .9oSPL. dyne2/cm 4
Figure 7. - Comparison of biased and unbiasednoncentral ehi-square distributions with an es-
timate obtained from synthesised data.
The results from one realization of this process are
shown in Figure 7 above. A pure tone of 20 Hzata level of 70 dB was added to noise at a level of 90
dB. The sample rate was 2344 per second so a data
record size of 2048 points gave a resolution of about
1.14 Hz and yielded an average noise level of about 60
dB in each frequency band. A 4-sample Kaiser-Bessel
ice = 3) window was used, two--record spectral averages
were calculated, and 1000 estimates at the frequencyband nearest the tone, 19.457 Hz, were recorded. The
histogram shows the approximate density function esti-mated from the synthesized data. The smooth curve
shown by the solid line is the noncentral chi-squaredensity function where the window bias correction was
made. The smooth curve shown by the dashed line
is the noncentral chi-square 'density function where nowindow bias correction was made.
The bias corrected theoretical curve agrees very wellwith the histogram estimate from synthesized data. A
parametric evaluation of the agreement between the-ory and synthesized data is summarized in Table 2 be-
low. The parameters of interest, mean, standard devi-
ation, skewness, and (excess) Kurtosis, are listed in thefirst column. The values for those parameters estimated
from the distribution of spectral estimates are listed in
the second column. The corresponding bias corrected
theoretical parameters appear in the third column. The
t statistic in the fourth column is for testing the null hy-pothesis that the estimated and theoretical parametersare equal. The critical value at the 50% level for 1000
samples is t.s0[s0s] = .6744 so the null hypothesis is notrejected for either the mean or the standard deviation.
The critical value at the 50% level for infinite degrees of
freedom is t.s0[oo ] : .6742 so the null hypothesis is also
not rejected for either the skewness or (excess) Kurtosis.
Table 2. - Selected parameters of biased noneen-
tral ehi-square distribution.
Parameter
Mean
Std. Dev.
Skewness
Kurtosis
Estimate
0.2182
0.08890.6224
0.4545
Theory t statistic0.2193 -.3791
0.0883 0.29100.6417 -.2495
0.5580 -.6698
Given the conflicting assumptions necessary to de-
rive the model and correct for bias, as well as limitations
imposed by the random number generator and the useof a discrete transform, it would seem that the non-
central chi-square distribution appropriately describes
the distribution of spectral estimates for a pure tone inGaussian noise and that the numerical evaluation of the
distribution is accurately accomplished.
Tiltrotor Hover Data
Acoustic data acquired during an XV-15 tiltrotor
hover test were obtained to evaluate the applicability
of the noncentral chi-square distribution to helicopterspectral estimates. Data from a single channel were
converted to engineering units and spectra were calcu-
lated from sequential segments of data with no overlapin the same manner as the synthesized data. Spectral
averageswere determined from the appropriate numberof spectral records and the resulting spectral estimate
at the frequency of greatest magnitude was recorded forstatistical analysis.
Because the noise level and tone level were not
known a priori, they were estimated from the spectra.
The sample rate for these data was 24590 per secondso a data record size of 8192 gave a resolution of about
3 Hz. A 4-sample Kaiser-Bessel (a = 3) window wasused, two-record spectral averages were calculated, and44 estimates were made from the limited amount of data
available. An average spectrum of all the estimates is
shown in Figure 8 below. The solid line is the averageof all of the spectra while the dashed lines describe
the envelope containing the 44 spectral averages of two
spectra each. The peak spectral level occurs at 27.02Hs and the background noise per band was estimated
from the spectral average curve to be about 56 dB at
that frequency. The tone level in that frequency band
was then found by subtracting the noise energy in the
band from the total energy in the band. No correctionwas made for bias because the spectral levels used for
this calculation were already biased.
tO0| j , , , I
-- Average
--- _'nvelope
9°r-t
J'Om 80-- I
(i.
U') 70 _60
ipi
50"" ' (o
, /
e
50
[stlmated Tone Level: 90.6
Estimated Frequency: 27.02 Hz I
Estimated Noise Level: 56.00 dB --I/
FFT size: 8192 |
Resolution: `3,00 Hz
Window: 4-sample Kalser-Ressel
Spectra: 2
_Sarnples: 44
L _ II i o
1O0 150 200
Frequency, Hz
Figure 8. - XV-15 tUtrotor spectral average and
envelope of spectra comprising the average.
The estimated tone and noise levels were used to
generate the noncentrai chi-square density function for
comparison with experimental data shown in Figure 9below. The histogram shows the approximate densityfunction estimated from the hover data. The smooth
curve shown by the solid line is the noncentral chi-
square density function based on estimated tone and
noise levels. The smooth curve shown by the dashed line
is the chi-square density function for the total energy
contained in the frequency band.
,50 z , j
_=).45C
-0_.4o
0.30
_.25
_ .20
'*-.15
O. 0
t I I I l t t I I
Estimated Tone Level: 90.61 dB --
Estimated Frequency: 27.02 Hz
Estimated Noise Level: 56.00 dB --
FF"T size: 8192 -Resolution: ,3.00 Hz
Window: 4-sample Kalsor-Bessel "
Spectra: 2
Samples: 44
-- Noncentral Chi-Square. X. z .._
--- Chi-Squaro. X¢ z
"- -I-----f_--d._. J. --,L_= __ J.__ l | l
50 I O0 150
SPL. dvne2/cm 4
Figure g. - Comparison ofnoncentraI chi-squareand ehi-square distributions with an estimate
obtained from XV-15 tiltrotor hover data.
The noncentral chi-square curve is much narrowerthan the histogram estimate from the hover data. On
the other hand, the chi-square curve is very much
broader than the histogram. While the histogramseems somewhat closer in appearance to the noncentral
theoretical distribution, the difference is striking when
compared to the excellent agreement with synthesiseddata.
An examination of assumptions used in deriving thetheoretical distribution should indicate the source of
disagreement between experiment and theory. Assump-tions made in deriving the theoretical function includerequirements that the noise be Gaussian and that the
tone, or the Fourier component of the periodic signal,
maintain constant frequency and amplitude. The hover-
ing tiltrotor was 100 feet above the ground at a horison-
tal distance of 500 feet from the microphone so propaga-
tion induced variability is an unlikely culprit. The high
tone-to-broadband ratio and the brevity of the test,30 seconds, make it unlikely that any non-Gaussian or
nonstationary character of the noise would explain the
disagreement. The blade passage frequency should be
stable so it seems that source amplitude variations may
account for the greater than expected variability. Ei-
ther the source level or the distribution of energy amongharmonics varied.
Helicopter Approach Data
Acoustic data acquired during a Sikorsky S-76 he-licopter approach were also obtained to evaluate the
applicability of the noncentral chi-square distribution
to helicopter spectral estimates. Data from a single
channel were converted to engineering units and spec-tra were calculated from segments of data in the same
manner as with the other data except for an overlap ofabout 35% between successive data segments. Spectral
10
overlapping was used to get as many independent esti-
mates as possible in a short time interval. Harris [12].has suggested that transforms with this level of over-
lap are essentially independent when good windows are
used. Spectral averages were determined from the ap-
p/0priate number of spectral records and the resulting
spectral estimate at the frequency of the second blade
passage harmonic was recorded for statistical analysis.Because the noise level and tone level were not
known a priori, they were estimated from the spectra.The sample rate for these data was 2344 per second so a
data record size of 2048 gave a resolution of about 1.14
Hs. A 4-sample Kaiser-Bessel (c_ = 3) window wasused, three-record spectral averages were calculated,and 35 estimates were made from a small fraction of
the available data. The helicopter was approaching at
a constant speed so the range was constantly shrinking
and the tone level constantly increasing. The datawere selected from a fairly short time interval at a
relatively long range to minimize the relative effect
of the changing range yet give sufficient data for ahistogram estimate of the density function.
An average spectrum of all the estimates from these
limited data is shown in Figure 10 below. The solid
line is the average of all of the spectra while the dashed
lines describe the envelope containing the 35 spectral
averages of three spectra each. The peak of the secondharmonic of the blade passage frequency occurs at 48.07
Hz and the background noise per band was estimated
from the spectral average curve to be about 58 dB at
that frequency. The tone level in that frequency band
was then found by subtracting the noise energy in theband from the total energy in the band. No correction
was made for bias because the spectral levels used forthis calculation were already biased.
tlO , , _ , [ J _ t _ l r _ _ _ [ j i--w J 1
--Average Estimated Tons Level: 85.04 dB
tO0 =- --- Envelope Estimated Frequency: 48.07Hz _j
Eslimaforl Noise Level: 58.00 dR J
FF'F slze: 2048 ]' Resolution: 1.14 Hz --1
SO _' Window: 4-sample Kalssr-Bessol ii=
i _ Spectra: 3 "
q
- 80 ; l.[ _, Samples: 35',I s, :i
i,, J_ ,
' h /",
SO " , ' "" ',' " ' , I.. ,, , _,;,, , , ,.'_,,.'_'_ ,, _ _* _ . , ,_,
, , _ ., ,,.,,, , ¢,.:,.,: -,_.,, :,,..,,500 ;A_'v,_,.:t-;i v_,,,x, ,AI^ _ ,,, ,v',:..
50 I00 t 50 200
Frequency, Hz
Figure 10. - 5-?6 helicopter spectral average and
envelope of spectra comprising the average.
The estimated tone and noise levels were used to
generate the noncentral chi-square density function for
comparison with experimental data shown in Figure11 below. The histogram shows the approximate den-sity function estimated from the approach data. The
smooth curve shown by the solid line is the noncentral
chi-square density function based on estimated tone
and noise levels. The smooth curve shown by the dashedline is the chi-square density function for the total en-
ergy contained in the frequency band.
The noncentral chi-square curve is much narrowerthan the histogram estimate from the hover data. The
chi-square curve shows much better agreement with thehistogram and would appear to correctly describe the
variability of the spectral estimates made from this data
set. The helicopter approached the microphone from
a considerable distance and it is virtually certain that
propagation induced variability plays a significant role
in the variability of the received signal. The frequency
stability of the source is probably good but the highly
directional nature of the sound from a helicopter rotor
and variations in aircraft attitude during forward flightat low altitude may cause variations in the sound level
emitted in the direction of the microphone. Examina-
tion of the envelope of spectra in Figure 10 reveals thattone variability was approximately the same as back-ground variability.
.90 ' I Il
.80 -
.70 -
E.6o -U .
_.50 --O "
"_ .40 --
C "
,_3.30 -
_ .20 -VI
1-
• '.10 -
0"0 10
l I I I I ' I ' I _-Estimated Tons Level: 85.04 dE --Estimated Frequency: 48.07 Hz
Estimated Noise Level: 58.00 dg --
FFT size: 2048
Resolution: 1.14 Hz --
Window: 4-sample Kalssr-Bssssl
Spectra: ,3 --
Samples: 35 .__
-- Noncentral Chi-Squars, Xe z "--- Chi-5quaro. Xo=
_"FFI---,-,-,:'__ , .-1 , _l , [ I20 30 40 50 60 70
SPL. dyne2/cm 4
Figure 11. - Comparison of noncentral ehi-
square and chi-square distributions with an es-
timate obtained from S-76 helicopter approachdata.
Summary and Conclusions
A probability density function for the variability
of ensemble average spectral estimates from helicopteracoustic signals in Gauuian background noise was eval-
uated. A brief summary of the noncentral chi-Kluare
and chi-square distributions and Gaussian approxima-
tions to them was presented. The details of the develop-
ment were presented in an appendix. Numerical meth-ods used to calculate the density function and determine
11
confidence limits were presented in another appendix.
A FORTRAN program that implements the numerical
methods to calculate confidence limits also appears inan appendix.
Examples were plotted to show differences and sim-
ilarities between the density functions. Plots were also
presented to show differences between confidence lim-
its versus the number of spectra included in an averagefor various confidence coefficients and tone-to-noise ra-
tios. The noncentral chi-square density function was
then compared with synthesized data that closely ap-
proximated assumptions made in development of the
model, with short range tiltrotor hover data, and with
very long range helicopter approach data.
The excellent agreement between synthesized dataand theoretical curves indicates that the numerical
methods and computer program worked as desired. Thesomewhat poorer agreement between the tiltrotor hover
data and theoretical curves is likely an indication thatthe assumption of constant source level was violated. In
this case the noncentral chi-square distribution was im-
perfect but showed better agreement with the data than
the chi-square distribution. The data from a helicopter
approaching an observer showed very poor agreementwith the noncentral chi-square distribution. The much
better agreement of this data with the chi-square den-
sity function is an indication that extremely variable
tone levels, whether from source variability or propaga-tion effects, completely invalidate the use of the noncen-
tral chi-square distribution. The noncentral chi-square
distribution should give excellent agreement, however,
over time scales where the tone levels do not vary sig-nificantly, as in wind tunnel measured acoustic data.
Acknowledgements
The author would like to thank Charlie Smith for
providing the S-76 helicopter approach data that was
used to calculate spectra for comparison with the non-central chi-square density function. He would also like
to thank Ken Rutledge for providing the XV-15 hover
data that was used for the same purpose, and for his
suggestions for improving this work.
References
1. Hardin, Jay C.: Introduction to Time Series
Analysis. NASA RP-1145, March 1986.
2. Bendat, Julius S. and Piersol, Allan G.: Random
Data: Analysis and Measurement Procedures, SecondEd. John Wiley & Sons, Inc., 1988.
3. Rutledge, Charles K.: On the Appropriateness
of Applying Chi-Square Distribution Based Confidence
Intervals to Spectral Estimates of Helicopter FlyoverData. NASA CR-181692, August 1988.
4. Davenport, Wilbur B., Jr. and Root, William L.:
An Introduction to the Theo_T o/Random Signals andNoise. McGraw-Hill Book Co., Inc., 1958.
5. Whalen, Anthony D.: Detection o/Signals inNoise Academic Press, Inc., 1971.
6. Burdic, William S.: Underwater Acoustic SystemAnalysis. Prentice-Hall, Inc., 1984.
7. Rice, S. O.: Statistical Froperties of a Sine Wave
Flus Random Noise. Bell Syst. Tech. Jour., Vol. 27,1948.
8. Reich, Edgar and Swerling, Peter: The Detection
o/a Sine Ware in Gaassian Noise. Journal of AppliedPhysics, Vol. 24, No. 3, March, 1953.
9. Hinich, Melvin J.: Detecting a Hidden Peri-odic Signal When Its Period is Unknown. IEEE Trans-
actions on Acoustics, Speech, and Signal Processing,Vol. ASSP-30, No. 5, October 1982.
10. Urick, Robert J.: Principles of Underloater
Sound, Third Ed. McGraw-Hill Book Co., Inc., 1983.11. Hoel, Paul G.: Introduction to Mathematical
Statistics, Fourth Ed. John Wiley & Sons, Inc., 1971.
12. Harris, F. J.: On the Use of Windows/or Har-monic Analysis with the Discrete Fourier _'anaform.
Proc. of the IEEE, Vol. 66, No. 1, 1978.
13. Gradshteyn, Izrail. S. and Ryshik, Iosif. M.:
Table of Integrals, Series, and FroducLg, Corrected and
Enlarged Ed. Academic Press, Inc., 1980.
14. Abramowits, Milton and Stegun, Irene A.:Handbook o/Mathematical Functions. Dover Publica-tions, Inc., 1970.
15. Davis, Philip J. and Rabinowits, Philip: Meth-ods of Numerical Integration, Second Ed. Academic
Press, Inc., 1984.
12
I
nn-0
c
C0
4
2
0
-2
-4
-6
-8
-100
I''''1''''1
Confidence Coefficient: 80.00%
I
i I
I
/
I
/
Tone-to-Noise Ratio
-co dB ..... Chi-Squore, X0=
0.0 dB _ Noncenfrol Chi-Squore, XR=5.0 dB o_o
10.0 dB
20.0 dB
5 10 15 20
Number of spectra25
Figure 12. - 80% confidence limits for various tone-to-nolse ratios and number o£ spectra.
I ,,,I,l,,I
Confidence Coefficient: 90.00_=
rn-0
amw
uc
-04--
C0
C_
Tone-to-Noise Ratio
-oo dB Chi-Square, X0z
0.0 dB --------- Noncentral Chi-Squore, XRz5.0 dB
10.0 dB
20.0 dB = _-
-150 5 10 15 20 25
Number of spectra
Figure 13. - 90% confidence limits for various tone-to-noise ratios and number of spectra.
, 13
Appendix A: Mathematical Model
Single Spectral Estimate
This appendix contains the derivation of the probability density function of a sample of the output of an energy
detection system with an input signal Consisting of a pure _bneandbroadband part along with broadband background
noise. This derivation continues from the body of the paper. The sinusoidal part of the signal may be written as
where P is a constant amplitude, wj is the constant frequency of the component of the periodic signal contained
within the pass band, and ¢ is a random variable distributed uniformly on (0, 2x). Over a time interval of length _',where 0 < t < r, the composite random process may be expressed as a Fourier series
oo
where the coefficients
am=- c/(t)cos _2 t dtT
'/:bm= r e/(t)sin dt
are Gaussian random variables that become uncorrelated as 7"increases without limit. The composite process maybe rewritten as
where
cl(t) = za(t)cos(w,t)- zb(t)sin(w,t)
Oo
m=1 I"
so that the filter Output may be written as the sum of the expressions for the sinusoidal part and the compositenarrow-band process. Thus,
where
z/(t) = X_(t) cos[a,,t] - Xb(t) sin[wot]
xo(t)= Pcos(¢)+ ..(t)
Xb(t) : Psin(_b) + Zb(t )
14
The random variables zat and zs¢ from za(t) and zs(_), respectively, can be shown [4] to be independent and
Gaussian with sero mean and a mean square of
where w is a dummy variable of integration, S(a_) is the spectral density, cr_ is the variance of the Composite narrow-
band random process, 0,_ is the variance of the narrow-band process associated with the signal, and cv_ is the
variance of the narrow-band process associated with the background noise. For consistency of notation the energy
associated with the sinusoidal part of the signal will be referred to as c,_ = P_/2. The joint probability density
function of the random variables zat and zbt is then
from which the joint probability density function of the random variables Xat, Xb,, and _b can be determined:
]If the filter output is expressed in terms of a sinusoidal function ofaJ, with an envelope A(t) and phase _(t) that
are slowly varying functions of time compared with cos(_,t):
then the envelope and phase may be written, respectively, as
A(f,) = [.X_(f,)q'- .X/_(I_)] 1/2
= tan-1LX.,(t)J
Since the only nonvanishing terms in the expressions for za(t) and Zb(t ) are those that fall in the narrow filter band,
the frequency components of the envelope and phase are confined to a similar band centered on zero frequency. The
joint probability density function of the envelope and phase random variables can be written as
At
p(Atl Otl _b) =14-'_2c l e-[A31+PI-2AIP c°I(@I-@)]II¢_
The probability density function of the envelope can be determined by integrating over _t and ¢ (with 0 = _t - @):
which can be written as
A,,) e_(A_+P,)l.ja._i ° f AtP_
15
where I00 is the zero-order modified Bessel function.
The output from the square-law detector is
• 3(t) = A_(0cos:%._+ ¢,(t)]= _a"(t)+ _A'(t)cosC2b.'._+ ¢'(0])
The integrator is assumed to be an ideal low-pass filter that passes all frequency components in the band without
distortion and rejects all other frequencies perfectly. The low-pass filter width is the same as the band-pass filter
width so that the square of the envelope function, with all its frequency components in the pass band, is passed
by the integrator. The high-frequency component of the square-law detector output, with a frequency of 2w,, is
rejected •because the band-pass filter width must be small compared with Wo to satisfy the assumption that c/(t)is a narrow-band Gaussian random process. The output from the integrator and, hence, from the energy detection
system is given by z(t) where
The probability density function of zt can be obtained by a simple transformation to give an expression
k _ J
which is referred to as a noncentral chi-square distribution [5]. This is the probability density function that describes
the variability inherent in a single spectral estimate when the time series contains a periodic part that is manifested
as a harmonic in the frequency band.
16
Appendix B: Numerical Methods
Density Function Evaluation
Although the density functions appear to be relatively clear and concise, their numerical evaluation is difficult.
When there is no periodic signal present the density function is the product of a constant, a power, and an exponential.
In general the function exhibits a region where the power component dominates, a region where the exponential
component dominates, and a region where the power and exponential components approximately balance. Because
the magnitudes of the individual components can exceed the capacity of a computer to easily represent them while
their product does not, the density function is evaluated numerically by expressing it in terms of an exponentialonly. Normalizing by the noise energy to simplify the expression gives
p(9) = elnN+(N-l)la(N_)-Nfl-ln(P(N))
where 17= Y/_'_. The Gamma function is calculated by an asymptotic approximation [14] when N is sufficiently
large. This expression is used even when a periodic signal is present if the ratio of tonal energy to broadband energyis less than 50 dB.
When a periodic signal of significant magnitude is present the inclusion of the modified Bessel function enormouslycomplicates the situation. Again normalizing by the noise energy to simplify the expression, the density functionwith a periodic signal is
N--_.__gl
There are a variety of equivalent forms for a modified Bessel function of integer order but they each present difficulties
for numerical evaluation. Four different forms, all from Abramowitz and Stegun [14], were chosen for evaluating theprobability density function. The first equation uses a polynomial approximation to I0 that has a different form foreach of two different regions
p(_) _ e-(_+a)[1 ÷ 3.5156229p 2 + 3.0899424p 4
+ 1.206749296 + 0.265973298
+ 0.0360768p I° + 0.0045813p Iz]
for p < 1 and
p(_) _ e-(_/_-_2[0.398942281 + 0.013285929 -I + 0.00225319p-_
÷ 0.00157565p-3 + 0.00916281p-4
- 0.02057706p-s + 0.02635537p-6
- 0.01647633p-7 + O.O0392377p-s]/44V/4V_
for p _> 1 where p = 8_/15.
The second equation uses a polynomial approximation to 11 that also has a different form for each of two differentregions
p(_) _ 8_e-2(9+a)[0.5 + 0.87890594p 2 + 0.51498869p 4
+0.15084934pa+0.02658733p s
+0.00301532pl°+0.00032411p;2]
for p < 1 and
17
p(_) ,_ _-/RZ e-2(-4_'-v_')'[0.398942281 _ 0.03988024p -1 _ 0.00362018p-2
+ 0.00163801p :_ - 0:0103]555p-4
+ 0.02282967p -s - 0.02895312p-8
+ 0.01787654p -7 _ 0.00420059p -s]
for p > 1 where p = 16_/7[_/15.
The third equation uses a uniform asymptotic expansion when both N > 2 and N + loglo(R ) > 3
P(17) _-, • -N(_+/l)+hx N+ Iz(ln:i-ln R)- ½In(2xL,)- tLhi (1-#-_2 )-#-i_-#-ln(1-#-_-'_:=1u_.(p)lv k)
where u = N - 1, !7 = 2(N/u)VF_, p = 1/_, and the parameter 17takes the simple form
1+ lv/i-.T , if
or the somewhat more complicated form
1 1 if !7 > 2,
and
1 P
Uk+l(p ) = _p'(1 - p2)u_(p) "t- 8io (1 - 5pi)uk(p)d p
where uo(p) = 1.
The fourth, and final, equation uses an ascending series representation of the modified Bessel function such that
K
v(#) _ _ e(Jv+2k)h_N+(N+k- I)h _+k L_R-N(_+R)-_ [r(k+or(h+N)]k=O
where K is determined when the K th term is su_ciently small compared with the sum.
Integration
A Gauss-Legendre numerical integration scheme takes the form
M
where the M weights, wj, and abscissas, uj, are found on the interval (-1, 1) by approximating the roots of Legendre
polynomials [15]. Because the probability density function sometimes exhibits significant values only over a finite
interval, a further approximation can be made by replacing the extreme limits of integration so that the confidence
limits are calculated by solving
_- ,,-,-,,j=l
18
and
j=l -
for L and D'. The lower limit of integration, L*, is arbitrarily set either to zero or to a value twelve standard
deviations below the mean, whichever is greater
L"= m_, (0,_- 12V_
where the normalized mean and variance are given by _ = 1 + R and Pz : (I + 2R)/N.
Root Location
Solving for the upper and lower confidence limits of a chi-square density function involves finding the root ofeach of two equations that exhibit the same mathematical form
N-I _k
o = f(O = ,7- e-_ _ T.'k=O
where r/= (1 + W)/2 for the lower limit and r/= (1 - W)/2 for the upper limit. Because both the first and second
derivative of this function exist in analytic form
f'(_) = e-' / _N-I
and
rapid convergence to the root can be achieved, given an acceptable initial guess, by using a refinement of Newton'smethod [15]
f(_) _,_i+_= ¢i - f,(_--_
with
f(_i) fu(_i)AI=I+
f'(_) 2f'(_)
where successive improvements are made to the approximate root until some convergence criterion is met. The
function need only be evaluated once at each step and additional advantages are gained by observing that fl(_) is
the last term in the summation in/(_) and also that only the ratio of f"(_) to ft(O must be calculated. The second
order correction term Ai can cause instability so, in practice, its value is restricted to the range (0.1, 2.0). Becausethe summation is an approximation to the exponential function
,v-_ 6k
k=O
19
the function f(_) is evaluated numerically by
N-1
---k=O
to minimize round-off difficulties.
An initial guess for determining the root is provided by an approximation to the inverse chi-square function
given by [14]
3
_Iv i- _-_ + _
where q is a rational approximation to the inverse standard normal distribution given by [14]
q=+(p_ 2"515517 + 0"802853p + 0"010328p2 )l + l.432788p + O.189269prt + O.OO1308p3
The sign of c; depends on whether the tail is to the right or left and
Because no simple expressions for the derivatives of the noncentral chi-square confidence limit functions exist, a
secant method is used in which the first derivative term of Newton's method is replaced by a secant approximation.The root of
j=l
which is the lower confidence limit, is found by making successive improvements to the approximate root
Li+l = Li + 6i
where
6,=-6,_, [ L') ]- y(L,_I)
until some convergence criterion is met. The correction term 5i can yield an approximate root outside tile range fi)_
which the density function is defined so, in practice, its value is restricted to the range (-.99Li. (_ - Li)/2). The
same method is used for the upper confidence limit using the appropriate function with the exception that there _lre
no restrictions to the range of 6 i.
An initial guess for the upper root is provided by the rational approximation to an inverse standard m-,rmal
distribution, ¢, given above translated and scaled by mean and variance equal to those of the noncentral chi squaredistribution
UI = _ ÷ qv"_-
20
An initial guess for the lower root is provided in nearly the same way when R is large, except that the value is notallowed to be too small
/'1 ----max(_/10,/_-I- c;V_)
The approximation to the inverse chi-square function, given above, is used when R is less than 1/10. The first
increment to the initial guess of the root_ 61, is arbitrarily set to/:_2/10.
21
Appendix C: FORTRAN Program
The FORTRAN program CHISQR calculates theoretical confidence limits for a chi-square distribution and a
noncentral chi-square distribution. An example of program output is appended.
22
program chisqr
C
C CHISOR calculates confidence limits for chi-square and noncentral
C chi-square probability density functions and compares them with
C Gaussian approximations
C
C .......................
C
integer no
parameter (no=S1)
real gla(no),glw(no),glx(no)
integer io,n
real aln,alo,arl,aru,csll,csul,del,eps,fn,fo,gcll,gcul
real gncll,gncul,nccsul,nccsll,sdn,snn,snr,snt
real splr,spn,tt,vv,w,xp,xlmax,xlmin
double precision dn,lnu,nu,snrl,snrd,ldn,aO,a1,1ngn
common /gla/ gla
common /glw/ glw
common /glx/ glx
common /dp/ n,snr,dn,lnu,nu,snrl,snrd,ldn,aO0al
common /Ingn/ ingn
real xion,glt
double precision Igamma
eps=l .e-6
sdn=12.
C
C .......................
C
C Calculate Gauss-Legendre integration weights and abscissas on (-1,1)
C
call gl(no,glx,glw)
do io=1+(no+l)/2,no
glx(io)=-glx(no+l-io)
glw(io)=glw(no+1-io)
enddo
C
C ........
C
C Enter the number of spectra, tone-to-noise ratio, and confidence level
C
write(S,699)
write(6,601) 'Enter the number of averages ' '
read(5,*) n
write(6,600)
write(6,601) 'Enter the tone-to-noise ratio (dB): '
read(5,*) splr
23
write(6,600)
write(6,601) 'Enter the confidence level (_)
read(5,*) w
C
C ....................................
C
C First guess from standard normal distribution inverse
C
aru=.5*(1.+.Ol*w)
arl=.5*(1.-.Ol*w)
tt=sqrt(-2.*alog(arl))
xp=tt-(2.515517+tt*(.802853+tt*.O10328))
& /(1.+tt*(1.432788+tt*(.189269+tt*.O01308)))
C
C ........
: J
C
C Calculate tone and noise values
C
snr=lO.**(.l*splr)
snrd=lO.dO**(.ldO*splr)
spn=l.+snr
C
C Standard deviations of normal distributions
C
snn=l./sqrt(float(n))
snt=sqrt((1.+2.*snr)/n)
C
C Calculate preliminary values for EDSPDF
C
dn=dble(n)
ldn=dlog(dn)aO=dn*(ldn-snrd)
nu=dn-l.dO
if (n.gt.l) lnu=dlog(nu)
snrl=dlog(snrd)
al=2.dO*Idn+snrl
Ingu=Igamma(n)
C
C Confidence limits for noise by chi-square distribution
C
csul=lO.*alog10(xion(arl,n))
csll=10.*aloglO(xion(aru,n))
C
C Confidence limits by Gausslan approximation
C
gncll=lO.*aloglO(amaxl(1.258925412e-lO,(1.-xp.snt/spn)))
gncul=lO.*aloglO(1.+xp*snt/spn)
gcll=10.*alog10(amaxl(1.25892S412e-lO,(1.-xp,snn)))
gcul=lO.*aloglO(1.+xp*snn)
C
C ..............
24
C
C Integration limits
C
xlmax=spn+sdn*snt
xlmin=amaxl(0.,spn-sdn*snt)
C
C ...................................
C
C
C
C
C
C ..........
Search for confidence limits of noncentral chi-square using a
modified secant method for the lower limit and a secant method
for the upper limit
First guess at lower limit
aln=amax1(.1_spn,spn-xp_snt)
First guess if chi-square
if (splr.le.-10.) then
vv=l./(9.*n)
aln=amaxl(O.,((1.-vv-xp*sqrt(vv))**3))
endif
C
C .........
C
C First function evaluation
C
fn=glt(xlmin,aln)-arl
del=(spn-aln)*.06
C
C .........
C
C Repeat until there is an answer
C
1 continue
alo=aln
aln=alo+del
C
C Next function evaluation
C
fo=fn
fn=glt(xlmin,aln)-arl
C
C Next delta
C
if (fo.eq.fn) goto 2
del=amax1(-.99*aln,amin1(.6*(spn-aln),del*fn/(fo-fn)))
if (abs(del).lt.aln*epe) goto 2
25
goto 1C
C Convergence
C
2 continue
nccsll=lO.*aloglO(aln/spn)
C
C ...................................
C
C First guess at upper limit
C
C
C
C
aln=spn+xp*snt
C First function evaluation
C
fn=glt(xlmin,aln)-aru
del=snt*.1
C
C .........
C
C Repeat until there is an answer
C
3 continue
alo=aln
aln=alo+del
C
C Next function evaluation
C
fo=fn
fn=glt(xlmin,aln)-aru
C
C Next delta
C
if (fo.eq.fn) goto 4
del=de!*fn/(fo-fn)
if (abs(del).it.aln*eps) goto 4
goto 3C
C Convergence
C
4 continue
nccsul=lO.*aloglO(aln/spn)
C
C
C
C Write results
C
write(6,602) 'Noise'
write(8,600)
26
erite(6,603)
write(6,603)
write(6,604)
write(6,604)
write(6,602)
write(6,600)
write(6,603)
write(6,603)
write(6,604)
write(6,604)
write(6,699)
stop ' '
' Chi-Square ,,, Gaussian
J
'Upper llmit',csul,' dB',Ecul ,, dB'
'Lower limit' cs11,' dB',Ec11 ,, d3'
'Tone + Noise, ',splr,' dB T/N Ratio'
'Noncentral Chi-Square ,,, Gaussian
'Upper Limit',nccsul,' dB',gncul,' dB'
'Lower Limit',nccsll,' dB,,Encll ,, dB'
C
C Formats
600 format(x)
601 format(x,a,$)
602 format(////,x,a,2x,f6.1,a)
603 format(16x,a,a)
604 format(x,a,3x,(Sx,f9.4,a,Sx),(Sx,f9.4,a))
699 format(/)
end
L
27
real function edspdf(z)
CC Noncentralchi-squared probability density function of order 2nC
C
real z
integer n
real snr
double precision dn,lnu,nu,snrl,snrd,ldn,aO0al,lnE n
common /dp/ n,snr,dn,lnu,nu,snrl,snrd,ldn,aO,al
common /lngn/ lngn
integer k
real qeta,qim,qin,qio,qip,qrho,qt,qti,qt2,qx,qzeta
double precision eps
double precision zz,zO
double precision bO,bl
double precision lnxi,lnxim,pdf,xi,lncf
double precision s
double precision zn,zs,eta,us,den,z2
double precision t,tt,ttt,tttt,ttttt
eps=l.d-16
C
C Evaluate PDF using polynomial approximation when N=I
C (Abramowitz and Stegunp 378)C
k
k
k
k
if (n.eq.1) then
if (z.eq.O.) then
edspdf=exp(-snr)
else
qeta=sqrt(snr)
qzeta=sqrt(z)
qx=2.*qeta*qzeta
qt=qx/3.75
if (qt.lt.1.) then
qt2=qt*qt
qio=l.
+qt2.(3.5156229
+qt2.(3.0899424
+qt2.(1.2067492
+qt2.(0.2659732
+qt2.(0.0360768
+qt2.(0.0045813))))))
edspdf=qio*exp(-(snr+z))
else
28
qti=l./qt
qim=0.39894228
+qti*(0.01328592
& +qti*(0.00225319
+qti*(-.00157585
• +qti*(0.00918281
• +qti*(-.02087706
• +qti*(0.02635537
• +qti*(-.01647633
• +qti*(O.OO392377))))))))
qrho=qzeta-qeta
edspdf=qim*exp(-qrho*qrho)/sqrt(qx)endif
endif
return
endif
C
C Evaluate PDF using polynomial approximation when N=2
C (Abramowitz and Stegunp 378)
if (n.eq.2) then
if (z.eq.O.) then
edspdf=O.
else
qeta=sqrt(snr)
qzeta=sqrt(z)
qx=4.*qeta*qzeta
qt=qx/3.75
if (qt.lt.1.) then
qt2=qt*qt
qip=0.5
• +qt2.(0.87890594
• +qt2*(O.S1498869
• +qt2.(0.15084934
• +qt2.(0.02658733
• +qt2.(0.00301632
• *qt2.(0.00032411))))))
edspdf=8.*z*qlp*exp(-2.*(snr+z))
else
qti=l./qt
qtn=0.39894228
• +qti*(-.03988024
• +qti*(-.00362018
• +qti*(0.00163801
• +qti*(-.01031885
• +qti*(0.02282967
• +qti*(-.02898312
& +qti*(0.01787684
29
& +qti*(-.00420059))))))))
qrho=qzeta-qeta
edspdf=sqrt(qzeta/(snr*qeta))*qin*exp(-2_,qrho,qrho)endif
endif
return
endif
C
C Evaluate PDF when N>2
C
if (z.eq.O.) then
if (n.gt.1) then
edspdf=O.
else
edspdf=exp(-snr)
endif
return
endif
zz=dble(z)
if (snrd.gt.l.d-5) then
if (dn+dloglO(s_rd).gt.3.dO) thenC
C Uniform asymptotic approximation (Abramowitz and Stegun p 378)C
zn=2.dO*(n/nu)*dsqrt(snrd*zz)
z2=zn*zn
if (zn.gt.2.dO) then
zs=zn*dsqrt(1.dO+l.dO/z2)
else
zs=dmaxl(1.dO,dsqrt(1.dO+z2))
endif
t=l.dO/zs
tt=t*t
ttt=t*tt
tttt=tt*tt
ttttt=tt*ttt
if (zn.gt.2.dO) then
eta=zs+dlog(l.dO/(l.dO/zn+dsqrt(l.dO+l.dO/z2)))
else
eta=zs+dlog(zn/(l.dO+dsqrt(l.dO+z2)))
endif
us=l.dO
den=nu
us=us+t*(.125dO
i +tt$(-.2083333333333333dO))/den
den=den*nu
us=us+tt*(.OTO3125dO
k +tt*(-.4010416666666667dO
+tt*(.3342013888888889dO)))/den
3O
den=den*nu
us=us+ttt*(.O73242187SdO
• +tt*(-.8912109375dO
• +tt*(1.846462673611111dO
• +tt*(-1.O25812S96450617dO))))/denden=den*nu
us=us+tttt*(.l1215209960937SdO
• +tt*(-2.3640869140625dO
• +tt*(8.78912353515625dO
• +tt*(-11.20700261622299dO
• +tt*(4.669584423426248dO)))))/den
den=den*nu
us=us+ttttt*(.2271080017089844dO
• +tt*(-7.368794359479632dO
• +tt*(42.53499874638846dO
• +tt*(-91.81824154324002dO
• +tt*(84.63621767460074dO
• +tt*(-28.21207255820025dO))))))/den
den=den*nu
us=us+ttt*ttt*(.S72SO14209747314dO
• +tt*(-26.49143048695155dO
• +tt*(218.1905117442116dO
• +tt*(-699.S796273761326dO
• +tt*(1059.990452528dO
• +tt*(-765.2S24681411817dO
• +tt*(212.STO1300392171dO)))))))/den
xl=dn*((nu/dn)*eta-(zz+snrd))
xi=xi+Idn-.SdO*dlog(zs)-.SdO,Inu
xl=xi+.SdO*nu*(dlog(zz)-snrl)
xi=xi-O.918938533204673dO
xi=xi+dlog(us)
edspdf=sngl(dexp(xi))
else
C
C Ascending series (Abramowitz and Stegunp 375)C
zO=dlog(zz)
bO=aO+nu*zO-dn*zz
bl=al+zO
pdf=O .dO
k=O
incf=lngn
Inxim=bO-Incf
1 continue
lnxi=bO+bl*k-lncf
if ((lnxt.lt.lnxim).and.(pdf.eq.O.dO)) goto 3
if (lnxt.lt.-3OO.dO) goto 2
xi=dexp(lnxi)
pdf=pdf+xi
if (xt.le.eps*pdf) goto 32 continue
31
k=k+l
Incl=Incf+dlog(dble(k))+dlog(dbls(n+k-1))
goto 1
3 continue
edspdf=sngl(pdf)
endif
else
C
C Broadband approximation _hen tone-to-broadband ratio is smallC
if (n.eq.1) then
edspdf=sngl(dn*dexp(-zz))else
if (z.eq.O.) then
edspdf=O.
else
s=zz_dn
edspdf=sngl(dexp(ldn+nu*dlog(s)-(s+lngn)))endif
endif
endif
return
end
32
doubleprecision function Igamma(n)
C
C Natural logaritDu, of the gamma function, G(n)
C
C ...........
C
C
C
integer n
integer i
double precision nn,in,ins
C
C Evaluate gamma function (Abramowitz and Stegun p 2S7)
C
if (n.lt.lO0) then
Igamma=O.
if (n.gt.2) then
do i=3,n
igamma=Igamma+dlog(i-l.dO)
enddo
endif
else
nn=dble(n)
in=l.dO/nn
ins=in*in
Igamma=(nn-.SdO)*dlog(nn)-nn
Igamma=Igamma+O.918938533204673dO
Igamma=Igamma+in*(+8.333333333333333d-2
+ins*(-2.777777777777778d-3
& +ins*(+7.936507936607937d-4
& +ins*(-5.9523809523809S2d-4))))
endif
return
end
_L
33
C
C
real function xion(a,n)
C
C
C
C Solve :
C
C
C
k
-x N-1 x
A -- e SUM ---
k--O k!
for x Eiven A and _, return x/N.
C
real a
integer n
double precision delt,eps,fxio,xin,xio,lfnk,d,xiol
real vv,t,xp
inteEer k
C
C Modified second order gegton's method _hen N>I (Davis and Rabinowitz
C p 114) (first estimate by Abramowitz and Stegun p 941 via p 933)C
xin=-dlog(dble(a))
if (n.gt.1) then
vv=l./(9.*n)
if (a.lt.O.5) then
t=sqrt(-2.*dlog(dble(a)))
xp=t-(2.515517+t*(.802853+t_.010328))
& /(1.+t*(1.432788+t*(.189269+t_.O01308)))
xin=n*(1.-vv+xp_sqrt(vv))_3
else
t=sqrt(-2._dlog(l.dO-a))
xp=t-(2.SiSSlT+t_(.802853+t*.O10328))
/(1.+t*(l.432788+t*(.189269+t*.O01308)))
xin=n_(1.-vv-xp*sqrt(vv))_,3
endif
eps=1.d-8
continue
xio=xin
xiol=dlog(xio)
fxio=dexp(-xio)
Ifnk=O.dO
do k=l,n-1
Ifnk=Ifnk+dlog(dble(k))
fxio=fxlo+dexp(k*xiol-lfnY-xlo)
enddo
delt=(fxio-a)*dexp(xio+lfnk+(1-n),xiol)
d=dmaxl(.IdO,dminl(2.dO,l.dO-delt,(.SdO,((n-i)/xio-l.dO))))
xin=xio+d_delt
34
real function glt (_aain,xmax)
C
C ....... -- .............
C
C Adjust Gauss-Legendre abcissas and integrate the PDF
C
C
C
C
integer no
parameter (no=51)
real gla(no),glw(no),glx(no)
common /gla/ gla
common Iglwl glw
common /glx/ glx
real xmin,xmax
integer io
real hr,ta
real edspdf
C
C Adjust Gauss-Legendre abcissas for interval (xmin,xmax)
C
hr=.5*(xmax-xmin)
t a=xmax+xmin
do io=1,(no+1)/2
gla(io)=xm±n+hr*(1.+glx(io))
enddo
do io=l+(no+l)/2,no
gla(io)=ta-gla(no+l-io)
enddo
C
C Integrate the PDF over (xmin,xmax)
C
glt=O.
do io=l,no
glt=glt+glw(io)*edspdf(gla(io))
enddo
glt=hr*glt
return
end
36
subroutinegl(n,x,w)
C
C Calculate Gauss-Legendre integration abscissas and weights on (-1,1)C
C
real x(1),e(1)
integer i,k,m,n
real den,dp,dpn,dl,d2pn,d3pn,d4pn,el
real fx,h,p,pk,pkml,pkpl,t,tl,u,v,xO
C
C Find Gauss-Legendre integration abscissas and weights on (-1,1)
C (Davis and Rabinowitz p 487)
C
m=(n+l)/2
e1=n* (n+ 1)do i=1,m
t= (4"i-1) *3. 1415926536/(4,n+2)
xO=(1.-(1.-1./n)/(8.,n,n,_),cos(t)
pkml=l.
pk=xO
do k=2,n
t l=xO*pk
pkp 1= t1-pkm 1- (t 1-pkm I)/k+t 1
pkm l=pk
pk =pkp 1
enddo
den= 1. -xO*xO
dl=n* (pkml-xO*pk)
dpn=dl/den
d2pn= (2. *xO*dpn-e l*pk)/den
d3pn= (4. *xO *d2pn+ (2. - •1 )*dpn )/den
d4pn= (6. *xO*d3pn+ (6. -el )*d2pn)/den
u=pk/dpn
v=d2pn/dpn
h=-u* (1. +. 5*u* (v+u* (v*v-d3pn/(3. *dpn) ) ) )
p=pk+h* (dpn+. 5*h* (d2pn+h/3. * (d3pn+. 25*h*d4pn) ) )
dp=dpn+h* (d2pn+. 5*h* (d3pn+h*d4pn/3.) )
h=h-p/dp
•(i)=xO+h
fx=dl-h*e 1" (pk+. 5*h* (dpn+h/3. * (d2pn+. 25.h* (d3pn+. 2*h*d4pn) )))
,(i)=2.*(1 .-xCi)*xCi))/Cfx*fx)
enddo
if (m+m.gt.n) x(m)=O.
return
end
37
Example run of FORTRAN program CHISQR:
Enter the number of averages : 5
Enter the tone-to-noise ratio (d3): 5
Enter the confidence level (_) : 80
Noise
Chi-Square Gaussian
Upper limit 2.0377 dB 1.9679 dB
Lower limit -3.1290 dB -3.6978 dB
Tone + Noise,
Upper Limit
Lower Limit
STOP
8.0 dB T/N Ratio
Noncentral Chi-Square
1.4144 dB
-1.9070 dB
Gaussian
1.3758 dB
-2.0253 dB
38
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1. AGENCY USE ONLY(Leave b/ank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED
October 1993 Contractor Report4. TITLE AND SUBTITLE
On the Use of the Noneentral Chi-Square Density Function for theDistribution of Helicopter Spectral Estimates
6. AUTHOR(S)Donald P. Garber
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
Lockheed Engineering & Sciences Company
144 Research Drive
Hampton, VA 23666-1339
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)
National Aeronautics and Space AdministrationLangley Research CenterHampton, VA 23681-0001
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C NASI-19000WU 505-63-70-02
8. PERFORMING ORGANIZATION
REPORT NUMBER
10. SPONSORING/MONITORINGAGENCY REPORT NUMBER
NASA CR-191546
Ill. SUPPLEMENTARY NOTES
Langley Technical Monitor: William L. Willshire, Jr.
12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE
Unclassified - Unlimited
Subject Category 71
13. ABSTRACT (Maximum 200 words)
A probability density function for the variability of ensemble averaged spectral estimates from helicopteracoustic signals in Gaussian background noise was evaluated. Numerical methods for calculating the densityfunction and for determining confidence limits were explored. Density functions were predicted for bothsynthesized and experimental data and compared with observed spectral estimate variability.
II I
14. SUBJECT TERMS
Spectralanalysis;Acoustics;Aircraftnoise
I
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